COMBINE syrup, butter, sugar, spices and salt in a medium saucepan over medium, stirring until butter is melted and mixture is smooth, 2 min. Let cool for 15 min.
WHISK flour and baking soda in a large bowl. Stir in syrup mixture, then egg, until blended. Divide in half and wrap in plastic wrap. Flatten into discs and refrigerate until firm, 1 hr.
PREHEAT oven to 325F. Line 2 baking sheets with parchment.
LIGHTLY flour a large piece of parchment. Roll 1 dough disc on parchment to 1/8-in. thickness. Cut out shapes using 2 1/2-in. to 4-in. cookie cutters, re-rolling scraps and refrigerating as needed. Arrange on prepared sheets and bake until edges are browned, 12 to 14 min. Let cool 5 min, then transfer to a rack to cool. Repeat with remaining dough.
ICING: Beat egg whites and vinegar in a large bowl with an electric mixer on medium until foamy. Beat in sugar until combined. Increase speed to medium-high and beat until thick, 2 min. Thin with water. Transfer to a piping bag fitted with a very small plain tip to decorate cooled cookies. Let sit at room temperature until icing hardens, about 4 hr.